If you want fame as a director, you've chosen the wrong medium. Even people with barely a passing interest in film would recognize the names Tarantino, Hitchcock, and Scorsese, but how many would know who Hal Prince or Susan Stroman or Michael Bennet are?
Sure, they may have heard of Bob Fosse, but it's probably in the context of Robin Williams dancing in The Birdcage. While directors are often the unsung heroes of any show they're involved in, at least they're famed within the industry. Also, if you direct a world premiere, your name will appear in the front of the published script for all time.
Of course, if your play gets really popular, you might get invited on Ellen and get introduced to the world of stay-at-home moms and dads. Maybe you'll be the next Julie Taymor, famous for The Lion King (one of the biggest successes in Broadway history) and Spider-Man: Turn Off the Dark (one of its biggest flops). Even fame doesn't mean everything's perfect.